 5.4.1: 1 through 7 deal with the massandspring system shown in Fig. 5.4....
 5.4.2: 1 through 7 deal with the massandspring system shown in Fig. 5.4....
 5.4.3: 1 through 7 deal with the massandspring system shown in Fig. 5.4....
 5.4.4: 1 through 7 deal with the massandspring system shown in Fig. 5.4....
 5.4.5: 1 through 7 deal with the massandspring system shown in Fig. 5.4....
 5.4.6: 1 through 7 deal with the massandspring system shown in Fig. 5.4....
 5.4.7: 1 through 7 deal with the massandspring system shown in Fig. 5.4....
 5.4.8: The massandspring system of 2, with F1.t / D 96 cos 5t, F2.t / 0
 5.4.9: The massandspring system of 3, with F1.t / 0, F2.t / D 120 cos 3t
 5.4.10: The massandspring system of 7, with F1.t / D 30 cost, F2.t / D 60...
 5.4.11: The massandspring system of 7, with F1.t / D 30 cost, F2.t / D 60...
 5.4.12: In 12 and 13, find the natural frequencies of the threemass system...
 5.4.13: In 12 and 13, find the natural frequencies of the threemass system...
 5.4.14: In the system of Fig. 5.4.12, assume that m1 D 1, k1 D 50, k2 D 10,...
 5.4.15: Suppose that m1 D 2, m2 D 1 2 , k1 D 75, k2 D 25, F0 D 100, and ! D...
 5.4.16: Figure 5.4.13 shows two railway cars with a buffer spring. We want ...
 5.4.17: If the two cars of both weigh 16 tons (so that m1 D m2 D 1000 (slug...
 5.4.18: If cars 1 and 2 weigh 8 and 16 tons, respectively, and k D 3000 lb=...
 5.4.19: If cars 1 and 2 weigh 24 and 8 tons, respectively, and k D 1500 lb=...
 5.4.20: 20 through 23 deal with the same system of three railway cars (same...
 5.4.21: 20 through 23 deal with the same system of three railway cars (same...
 5.4.22: 20 through 23 deal with the same system of three railway cars (same...
 5.4.23: 20 through 23 deal with the same system of three railway cars (same...
 5.4.24: . In the threerailwaycar system of Fig. 5.4.6, suppose that cars ...
 5.4.25: Suppose that m D 75 slugs (the car weighs 2400 lb), L1 D 7 ft, L2 D...
 5.4.26: Suppose that k1 D k2 D k and L1 D L2 D 1 2L in Fig. 5.4.14 (the sym...
 5.4.27: In 27 through 29, the system of Fig. 5.4.14 is taken as a model for...
 5.4.28: In 27 through 29, the system of Fig. 5.4.14 is taken as a model for...
 5.4.29: In 27 through 29, the system of Fig. 5.4.14 is taken as a model for...
Solutions for Chapter 5.4: SecondOrder Systems and Mechanical Applications
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 5.4: SecondOrder Systems and Mechanical Applications
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. Chapter 5.4: SecondOrder Systems and Mechanical Applications includes 29 full stepbystep solutions. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This expansive textbook survival guide covers the following chapters and their solutions. Since 29 problems in chapter 5.4: SecondOrder Systems and Mechanical Applications have been answered, more than 15832 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.